What a pain

Algebra Level 2

Let α \alpha and β \beta denote the positive and negative value of x x respectively, where x x satisfy the equation 3 x 2 − x + 1 = 27 3^{x^2-x+1} = 27 , find α β + β α β \dfrac{\alpha^\beta + \beta}{\alpha \beta} .


The answer is 0.25.

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Russel Ryan Floresca by Russel Ryan Floresca , 20, Philippines

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1 solution

Akshat Sharda
Jan 31, 2016

3 x 2 − x + 1 = 27 x 2 − x + 1 = 3 ( x − 2 ) ( x + 1 ) = 0 x = 2 ⏟ α , − 1 ⏟ β ⇒ α β + β α β = 2 − 1 − 1 2 ( − 1 ) = 1 4 = 0.25 \begin{aligned} 3^{x^2-x+1} & = 27 \\ x^2-x+1 & = 3 \\ (x-2)(x+1) & = 0 \\ x & = \underbrace{2}_{\alpha},\underbrace{-1}_{\beta} \\ \Rightarrow \frac{\alpha^\beta+\beta}{\alpha \beta} & = \frac{2^{-1}-1}{2(-1)} \\ & = \frac{1}{4}=\boxed{0.25} \end{aligned}

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Wow! Very neat solution!! =D

Pi Han Goh - 5 years, 4 months ago

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